Problem: Integrate. $ \int 3\csc^2(x)\,dx $ Choose 1 answer: Choose 1 answer: (Choice A) A $3\sec(x)+C$ (Choice B) B $-3\sec(x)+C$ (Choice C) C $-3\cot(x)+C$ (Choice D) D $3\cot(x)+C$
We need a function whose derivative is $3\csc^2(x)$. We know that the derivative of $\cot(x)$ is $-\csc^2(x)$, so let's start there: $\dfrac{d}{dx} \cot(x) = -\csc^2(x)$ Now let's multiply by $-3$ : $\dfrac{d}{dx} \left[ -3\cot(x) \right]= -3\dfrac{d}{dx} \cot(x) = 3\csc^2(x)$ Because finding the integral is the opposite of taking the derivative, this means that: $ \int 3\csc^2(x)\,dx =-3 \cot(x)\, + C$ The answer: $-3 \cot(x)\, + C$